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<b>let </b><font color="Maroon">L</font> be   <a href="lattices.html#NM1">Lattice</a>;<br/><b>let </b><font color="Maroon">I</font> be    <a href="filter_2.html#M2" target="_self">Ideal</a> of <font color="Maroon">L</font>;<br/><b>let </b><font color="Maroon">J</font> be    <a href="filter_2.html#M2" target="_self">Ideal</a> of <font color="Maroon">L</font>;<br/>





<a NAME="E1:73"/>
( <span class="p1"><a href="filter_2.html#K19" target="_self">(.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">J</font></span>)</span></span><a href="filter_2.html#K19" target="_self">.&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="filter_2.html#K3" target="_self">&lt;.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">I</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">J</font></span>)</span> <a href="filter_2.html#K7" target="_self">.:</a> </span>)</span></span><a href="filter_2.html#K3" target="_self">.)</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">J</font></span>)</span> <a href="filter_2.html#K7" target="_self">.:</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">J</font> &amp; <font color="Maroon">I</font> <a href="filter_2.html#K15" target="_self">.:</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> &amp; <font color="Maroon">J</font> <a href="filter_2.html#K15" target="_self">.:</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">J</font> )
 
<b>by </b><i><a class="txt" href="filter_2.html#E8:73"><i><font color="Green">E22</font></i></a></i>;<br/>
<a NAME="E2:73"/><b>then </b><i><font color="Green">E14</font></i>: 
<span class="p1"><a href="filter_2.html#K19" target="_self">(.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">J</font></span>)</span></span><a href="filter_2.html#K19" target="_self">.&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">r'</font> where <font color="Olive">r'</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="lattice2.html#K1">.:</a> </span>)</span> :  ex <font color="Olive">p'</font>, <font color="Olive">q'</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="lattice2.html#K1">.:</a> </span>)</span> st <br/>( <font color="Olive">p'</font> <a href="lattices.html#K4">"/\"</a> <font color="Olive">q'</font> <a href="lattices.html#R3">[=</a> <font color="Olive">r'</font> &amp; <font color="Olive">p'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <font color="Olive">q'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="filter_0.html#T48">FILTER_0:48</a></i>;<br/>
<b>thus </b><a NAME="E3:73"/>
<span class="p1"><a href="filter_2.html#K19" target="_self">(.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">J</font></span>)</span></span><a href="filter_2.html#K19" target="_self">.&gt;</a></span> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> :  ex <font color="Olive">p</font>, <font color="Olive">q</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">r</font> <a href="lattices.html#R3">[=</a> <font color="Olive">p</font> <a href="lattices.html#K3">"\/"</a> <font color="Olive">q</font> &amp; <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> ) </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:73_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="filter_2.html#K19" target="_self">(.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">J</font></span>)</span></span><a href="filter_2.html#K19" target="_self">.&gt;</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">r'</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="lattice2.html#K1">.:</a> </span>)</span><b> such that </b><br/><a NAME="E3:73_1"/><i><font color="Green">E15</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r'</font> &amp;  ex <font color="Olive">p'</font>, <font color="Olive">q'</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="lattice2.html#K1">.:</a> </span>)</span> st <br/>( <font color="Olive">p'</font> <a href="lattices.html#K4">"/\"</a> <font color="Olive">q'</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">r'</font> &amp; <font color="Olive">p'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <font color="Olive">q'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> ) )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">p'</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="lattice2.html#K1">.:</a> </span>)</span>, <font color="Maroon">q'</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="lattice2.html#K1">.:</a> </span>)</span><b> such that </b><br/><a NAME="E5:73_1"/><i><font color="Green">E16</font></i>: 
( <font color="Maroon">p'</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">q'</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">r'</font> &amp; <font color="Maroon">p'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <font color="Maroon">q'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> )
 <b>by </b><i/>;<br/>
<a NAME="E6:73_1"/>
(  <a href="filter_2.html#K6" target="_self">.:</a> <font color="Maroon">r'</font> <a href="lattices.html#R3">[=</a>  <a href="filter_2.html#K6" target="_self">.:</a> <span class="p1">(<span class="default"><font color="Maroon">p'</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">q'</font></span>)</span> &amp;  <a href="filter_2.html#K6" target="_self">.:</a> <span class="p1">(<span class="default"><font color="Maroon">p'</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">q'</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p'</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">q'</font> &amp; <font color="Maroon">p'</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">q'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="filter_2.html#K6" target="_self">.:</a> <font color="Maroon">p'</font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><a href="filter_2.html#K6" target="_self">.:</a> <font color="Maroon">q'</font></span>)</span> &amp;  <a href="filter_2.html#K6" target="_self">.:</a> <font color="Maroon">p'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'</font> &amp;  <a href="filter_2.html#K6" target="_self">.:</a> <font color="Maroon">q'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q'</font> &amp;  <a href="filter_2.html#K6" target="_self">.:</a> <font color="Maroon">r'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r'</font> )
 
<b>by </b><i><a class="ref" href="filter_2.html#T3" target="_self">Th3</a>, </i>;<br/>
<b>hence </b><a NAME="E7:73_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> :  ex <font color="Olive">p</font>, <font color="Olive">q</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">r</font> <a href="lattices.html#R3">[=</a> <font color="Olive">p</font> <a href="lattices.html#K3">"\/"</a> <font color="Olive">q</font> &amp; <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> ) </span>}</span> 
 <b>by </b><i>, <a class="ref" href="filter_2.html#T3" target="_self">Th3</a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E4:73"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> :  ex <font color="Olive">p</font>, <font color="Olive">q</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">r</font> <a href="lattices.html#R3">[=</a> <font color="Olive">p</font> <a href="lattices.html#K3">"\/"</a> <font color="Olive">q</font> &amp; <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">r</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E6:73"/><i><font color="Green">E21</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> &amp;  ex <font color="Olive">p</font>, <font color="Olive">q</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Maroon">r</font> <a href="lattices.html#R3">[=</a> <font color="Olive">p</font> <a href="lattices.html#K3">"\/"</a> <font color="Olive">q</font> &amp; <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> ) )
 ;<br/>
<b>consider </b><font color="Maroon">p</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">q</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E8:73"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">r</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">p</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">q</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> )
 <b>by </b><i/>;<br/>
<a NAME="E9:73"/>
( <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">q</font></span>)</span> <a href="filter_2.html#K5" target="_self">.:</a>  <a href="lattices.html#R3">[=</a> <font color="Maroon">r</font> <a href="filter_2.html#K5" target="_self">.:</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">q</font></span>)</span> <a href="filter_2.html#K5" target="_self">.:</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">q</font> &amp; <font color="Maroon">p</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="filter_2.html#K5" target="_self">.:</a> </span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="filter_2.html#K5" target="_self">.:</a> </span>)</span> &amp; <font color="Maroon">p</font> <a href="filter_2.html#K5" target="_self">.:</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp; <font color="Maroon">q</font> <a href="filter_2.html#K5" target="_self">.:</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> &amp; <font color="Maroon">r</font> <a href="filter_2.html#K5" target="_self">.:</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> )
 
<b>by </b><i><a class="ref" href="filter_2.html#T3" target="_self">Th3</a>, </i>;<br/>
<b>hence </b><a NAME="E10:73"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="filter_2.html#K19" target="_self">(.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">J</font></span>)</span></span><a href="filter_2.html#K19" target="_self">.&gt;</a></span>
 <b>by </b><i>, , <a class="ref" href="filter_2.html#T3" target="_self">Th3</a></i>;<br/>


</div>
